A Tale of Three Waves

A brief introduction to oceanic waves April 16, 2015 Daniel Whitt Department of Applied Mathematics and Theoretical Physics University of Cambridge [email protected] http://www.damtp.cam.ac.uk/user/dw494 1 Introduction: A tale of three waves The first of two lectures to be delivered at Capital Normal University, Beijing March 30-April 3, 2015 Many textbooks treat oceanic wave motion extensively (e.g. Phillips, 1966; Leblond and Mysak, 1978; Lighthill, 1978; Gill, 1982; Pedlosky, 1986; 2003) and the ideas have been studied for many centuries (e.g. the tidal equations were first derived by Pierre-Simon Laplace in 1776). As a result, I can't possibly give either a particularly original or comprehensive treatment of oceanic wave motions in this first lecture. Previous researchers and teachers deserve much (if not all) of the credit for what appears here. Rather, my goal is to give a descriptive introduction to oceanic wave motions using both ob- servations and mathematical analysis. My hope is that after this lecture you will have some appreciation for 1) the ubiquity and diversity of wave phenomena in the ocean, and 2) the prin- ciple mathematical models and techniques that have been used to interpret and understand these motions in atmospheric science, oceanography and geophysical fluid dynamics. In the second lecture, I will build on these ideas by discussing some topics in ocean wave physics that are \special" in that I have worked on them and/or the topic particularly interests me. 1.1 Introduction The ocean can support a variety of classes of waves (which we loosely define as traveling perturbations to an equilibrium background state with a propagation speed that is typically different from the background flow speed). These classes are distinguished by the dominant restoring forces to a given departure from the equilibrium, including e.g. frictional, gravitational, pressure, and inertial forces (we entirely ignore negligible electromagnetic forces in the dynamics of Earth's ocean). Strictly speaking, all of the possible forces tend to be acting to some degree at all spatio-temporal scales. However, there is often a great deal to be learned about the dynamics of the ocean by isolating 1 the mechanics associated with the dominant restoring force (or forces) at a particular scale (time and space or, equivalently, frequency and wavelength) and investigating the physics of a reduced physical and mathematical model. Figure 1: Schematic of the energy spectrum of ocean variability. From Leblond and Mysak (1978) Waves in the ocean. This is because the governing physical model (the Navier-Stokes system) describes an extreme diversity of phenomena; characteristic length scales range from thousands of km (such as those associated with planetary waves and synoptic weather patterns in the atmosphere) to mm (such as capillary waves in a lake) and time scales range from seconds to months (see Figure 1).1 Both the time and space scales of typical wave motions range over more than 7 orders of magnitude in the ocean (see e.g. Figure 1). Even if analytic solutions to the Navier Stokes equations were always available, a detailed investigation of the properties of the different solutions would still be necessary to understand the practical implications of the different solutions (of course analytic solutions are also extremely hard to obtain and existence and uniqueness proofs are arguably among the most important unsolved problems in all mathematics). 1.2 Equations of Motion As I already suggested, the governing mathematical model for all the wave phenomena to be discussed is the rotating, incompressible, Navier-Stokes system. In the context of Earth, these equations are best presented in spherical coordinates. However, here we will consider motions that can be studied using local tangent-plane geometries where Cartesian coordinates are appropriate 1At the outset, we neglect sound waves, which require a compressible set of equations and have a negligible impact on the dynamics. 2 Figure 2: A schematic of our coordinate system, from Vallis's Atmospheric and Oceanic Fluid Dyn. textbook. (see Figure 2). Moreover, because the density varies by no more than about 1% from the background 3 value of ρ0 ≈ 1025 kg/m , we include the local density (as opposed to the reference density) only when multiplied by gravity. This simplification is known as the Boussinesq approximation. These equations of motion can be written: @u rp + u · ru + f × u = − + b + ν∆u ; (1) @t | {z } | {z } ρ0 |{z} |{z} |{z} Advection Coriolis |{z} Buoyancy Friction Time tendency Pressure Gradient @b + u · rb = κ∆b; (2) @t r · u = 0; (3) The r is the three-dimensional gradient operator, ∆ = r · r is the three-dimensional Laplacian operator, × denotes the vector cross-product and ∇· denotes the three-dimensional divergence operator. The three-dimensional velocity vector field u = (u; v; w) (in units of m/s) points locally to the east, north and upwards respectively in the direction of unit vectors (i; j; k) (see Figure 2). The buoyancy force b = (0; 0; b) defines the local vertical, where b = −gρ/ρ0. In the ocean, the density ρ (in units of kg/m3) depends non-linearly on two tracers (temperature and salinity), which diffuse at different rates. Moreover, the density also depends weakly on pressure. In this presentation, I will ignore these subtleties and assume that the density can be modeled as a single tracer with a single molecular diffusivity κ (in units of m2/s with the same magnitude as the kinematic viscosity ν), 3 e.g. as if the density depends linearly on temperature. The gravity g includes the true gravitational acceleration modified by a small centrifugal acceleration (the modification to g = 9:81 m/s2 due to the centrifugal force is of the order of 0.3% so it is safe to ignore it here). The Coriolis force reduces to f × u = (−fv; fu; 0) because we keep only the local vertical component of the Coriolis vector so −5 −1 that f = (0; 0; 2Ωe sin(θ)) where Ωe ≈ 7:3 × 10 s is the angular rotation rate of Earth and θ is the local latitude (see Figure 2). When I consider large scale Rossby wave motions, I will define a −4 beta plane where f = f0 + βy is a linear function of the y coordinate. Otherwise, f = f0 ≈ 10 s−1 in the mid-latitudes.2 Now we will derive governing equations and wave solutions for three different classes of wave motion, with characteristic horizontal scales ranging from small (∼ 1 − 100 m) to large (∼ 100 km) and timescales ranging from short (a few seconds) to long (at least a few days). The three classes are: 1) waves at the air/water interface, 2) internal inertia-gravity waves in a rotating, density-stratified fluid, 3) Rossby waves in a stratified fluid with a variable rotation rate (i.e. on a beta-plane). 1.3 Example 1: Surface gravity waves Probably the most familiar class of ocean wave is the surface gravity wave at the air-water interface. These motions are characterized by periods of a few seconds, much shorter than the rotational period of Earth (see Figure 1). They can be generated locally by winds applied over a short \fetch" or they may propagate in from afar as \swell" generated by a large storm. Figure 3 shows an image of surface waves shoaling at a beach. Figure 3: A photograph of surface gravity waves shoaling as they approach a beach. From http://www.blog.buoyalarm.com/how-to-read-ocean-buoys-tips-resources. Here I will introduce a simple model for wave propagation (that does not describe the generation or dissipation process) based on several simplifications of the governing equations (1). Suppose the wave motion has a characteristic time scale T~ = 2π=!~, length scale L~ = 2π=k~ (both horizontal and vertical), and velocity scale U~, where! ~ is the characteristic scale for the wave frequency and k~ is the characteristic scale for the horizontal wave number (in general, tildes will denote characteristic 2See e.g. Vallis, (2006) Atmospheric and Oceanic Fluid Dynamics or Phillips (1966), The dynamics of the upper ocean or any of a number of similar textbooks for more on the equations of motion. 4 scales rather than the dynamical variables). Note the crucial implicit assumption that the aspect ratio of the motion is order one, that is the vertical length scale and velocity scale is equivalent to the horizontal length and velocity scale, U~ ∼ V~ ∼ W;~ (4) L~ ∼ H;~ (5) and that the pressure scales with the acceleration term P~ ∼ U~!=~ k~. Then we apply the following explicit assumptions: 1. The wave frequency is much larger than the Coriolis frequency so that the time tendency is much greater than the Coriolis force in (1). That is, f << 1: (6) !~ 2. The wave has a sufficiently small characteristic velocity U that the advective terms are much smaller than the time tendency in (1). That is, U~k~ << 1: (7) !~ 3. The wave motion is not affected by friction; the time tendency is much larger than the dissipative term. That is, νk~2 << 1: (8) !~ In any all three cases the significance of the respective forces has been expressed as a ratio of two time scales or, equivalently, a ratio of two frequencies. As an aside, note that the assumption of an order-one aspect ratio and a characteristic timescale that is much shorter than the advective timescale implies that density perturbations have a negligible effect on gravity wave motion and the fluid can be treated as if it has a uniform density. Incidentally, this implies that the motion has a frequency that is much larger than the local buoyancy frequency, r s @b −g @ρ ! >> N = = : (9) @z ρ0 @z In general, one should consider carefully the limitations of the different assumptions.

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